The closest I could find was System.Net.Mime.MediaTypeNames , but it doesn't seem to have everything (like json), as it seems to be more focused on email attachments.
System.Net.Mime.MediaTypeNames
Enumeration does not make much sense. MIME types are public. That is, the list is not finite: new types are added from time to time.
See RFC4288: Media Type Specifications and Registration Procedures
The IANA database is likely to be complete, but you will need to analyze these pages to get a flat list.
There is also a mime.types file that ships with Apache, which appears to have been obtained from the specified list.